Solve for $x$ and $y$ using elimination. $\begin{align*}-4x+4y &= -7 \\ 5x-6y &= 7\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $2$ $\begin{align*}-12x+12y &= -21\\ 10x-12y &= 14\end{align*}$ Add the top and bottom equations. $-2x = -7$ Divide both sides by $-2$ and reduce as necessary. $x = \dfrac{7}{2}$ Substitute $\dfrac{7}{2}$ for $x$ in the top equation. $-4( \dfrac{7}{2})+4y = -7$ $-14+4y = -7$ $4y = 7$ $y = \dfrac{7}{4}$ The solution is $\enspace x = \dfrac{7}{2}, \enspace y = \dfrac{7}{4}$.